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Question
A die is thrown thrice. Find the probability of getting an odd number at least once.
Solution
\[P\left( \text{ getting an odd number in one throw } \right) = \frac{1}{2}\]
\[\text{ Here, getting an odd number in three throws refers to 3 independent events } .\]
\[P\left( A \right) = P\left( B \right) = P\left( C \right) = \frac{1}{2}\]
\[P\left( A \cup B \cup C \right) = P\left( A \right) + P\left( B \right) + P\left( C \right) - \left[ P\left( A \cap B \right) + \left( B \cap C \right) + \left( C \cap A \right) \right] + P\left( A \cap B \cap C \cap \right)\]
\[ = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} - \left[ \frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2} \right] + \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\]
\[ = \frac{3}{2} - \frac{3}{4} + \frac{1}{8}\]
\[ = \frac{12 - 6 + 1}{8}\]
\[ = \frac{7}{8}\]
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